204 Proceedings of the Royal ISociety of Victoria. 



2"". — With respect to all the lines /j /j situated in tangent 

 planes to the quadric Qi . 



We may first observe that if Pj Pj 1\ be any fixed plane 

 tangent to the quadric Q^, and that we project the quadric 

 itself orthogonally by means of othe)- tangent planes upon 

 J\ Pi Pi, then will the ])rojection be a conic c^ q c^ situated 

 in the plane PiP^P^, which is obviousl}^ the envelope of 

 all the lines li /^ in the plane. 



3°. — With respect to all tlie lines l^ li situated in any 

 plane B BB whatever. 



We first proceed and find the sum s,^ of the moments of 

 inertia of the entities a^.i/i, a.,. M2, a.^. M^,, . . . , with 

 respect to the plane BBS. We then find the quadric Q^ 

 such that the sum of the moments of inertia of the entities 

 with respect to any of its tangent planes is = 2.Si, — s^ . Then, 

 ol)viously, the orthogonal projection of the quadric Q^ so 

 found (by means of tangent [)lanes to it) upon the plane 

 B BB will be a conic, which is the envelope of the lines l^ l^ 

 situated in the plane. 



The foliowins^ is an obvious deduction : — 



Theorem .'>. 



Given any masses M^, M2, M-^, ... in space, and corres- 

 ponding numbers a^, a.^, «3, . . . of known signs as multi- 

 pliers ; and given also the system of confocal quadrics 

 Qi, Q'^' Qz> • • • ■> such that the sum of the moments of 

 inertia of the entities a^. ili,, a.-M.^, a-^. 31-^, . . . , with 

 respect to tangent planes to the quadrics are equals 

 respectively to 6'i, s.,, .^.g, . . . ; then the orthogonal pro- 

 jections of the quadrics on any given plane B B B in space, 

 constitute a family of confocal conies, which are the 

 respective envelopes of straight lines l^li, l.^l-i, Izh, • . • ■ , 

 situated in the plane, such that the sum of the moments of 

 inertia of the entities ftj. i/j, a._,. M-2> (-h- ^Z' • • • > with 

 ]-espect to them, are determinable constants. And if the 

 plane P^ B^ B be parallel to either one of the two systems of 

 parallel circular sections of the conibcal quadrics, then will 



